Spin and Magnetic Moments

1. Spin and Magnetic Moments • Orbital and intrinsic (spin) angular momentum produce magnetic moments • coupling between moments shift atomic energies • Look first at orbital (think of current in a loop) • the “g-factor” is 1 for orbital moments. The Bohr magneton is introduced as natural unit and the “-” sign is due to the electron’s charge P460 - Spin

2. Spin • Particles have an intrinsic angular momentum - called spin though nothing is “spinning” • probably a more fundamental quantity than mass • integer spin --> Bosons half-integer--> Fermions • Spin particle postulated particle • 0 pion Higgs, selectron • 1/2 electron photino (neutralino) • 1 photon • 3/2 delta • 2 graviton • relativistic QM uses Klein-Gordon and Dirac equations for spin 0 and 1/2. • Solve by substituting operators for E,p. The Dirac equation ends up with magnetic moment terms and an extra degree of freedom (the spin) P460 - Spin

3. Spin 1/2 expectation values • similar eigenvalues as orbital angular momentum (but SU(2)) • Dirac equation gives g-factor of 2 • non-diagonal components (x,y) aren’t zero. Just indeterminate. Can sometimes use Pauli spin matrices to make calculations easier • with two eigenstates (eigenspinors) P460 - Spin

4. Zeeman Effect • Angular momentum->magnetic moment->energy shifts • additional terms in S.E. do spin-orbit later. Right now assume atom in external magnetic field • look at ground state of H. L=0, S=1/2 P460 - Spin

5. Spin 1/2 expectation values • Let’s assume state in a combination of spin-up and spin-down states (it isn’t polarized). • Can calculate some expectation values. Griffiths Ex. 4.2. Z-component • x-component • as normalized, by inspection P460 - Spin

6. Griffiths Prob. 4.28. For the most general normalized spinor find expectation values: • just did x and z • repeat for other • note x and y component will have non-zero “width” for their distributions as not diagonalized P460 - Spin

7. Widths • Can look at the widths of spin terms if in a given eigenstate • z picked as diagonal and so • for off-diagonal P460 - Spin

8. Components, directions, precession • Assume in a given eigenstate • the direction of the total spin can’t be in the same direction as the z-component (also true for l>0) • Example: external magnetic field. Added energy • puts electron in the +state. There is now a torque • which causes a precession about the “z-axis” (defined by the magnetic field) with Larmor frequency of z B S P460 - Spin

9. Angles • Griffiths does a nice derivation of Larmor precession but at the 560 level • to understand need to solve problem 4.30. • Construct the matrix representing the component of spin angular momentum along an arbitrary radial direction r. Find the eigenvalues and eigenspinors. • Put components into Pauli spin matrices • and solve for its eigenvalues P460 - Spin

10. Go ahead and solve for eigenspinors. • Phi phase is arbitrary. gives • if r in z,x,y-directions P460 - Spin

11. Combining Angular Momentum • If have two or more angular momentum, the combination is also eigenstate(s) of angular momentum. Group theory gives the rules: • representations of angular momentum have 2 quantum numbers: • combining angular momentum A+B+C…gives new states G+H+I….each of which satisfies “2 quantum number and number of states” rules • trivial example. Let J= total angular momentm P460 - Spin

12. Combining Angular Momentum • Non-trivial example. • Get maximum J by maximum of L+S. Then all possible combinations of J (going down by 1) to get to minimum value |L-S| • number of states when combined equals number in each state “times” each other • the final states will be combinations of initial states. The “coefficiants” (how they are made from the initial states) can be fairly easily determined using group theory (and step-up and step-down operaters). Called Clebsch-Gordon coefficients P460 - Spin

13. 2 terms • Same example. • Example of how states “add”: • Note Clebsch-Gordon coefficients P460 - Spin

14. Clebsch-Gordon coefficients for different J,L,S P460 - Spin